Uniform cylinder of density $\rho$  and cross -sectional area A floats in equilibrium in two non- mixing liquids of densities ${\rho }_1\text{and}{\rho }_2$ as shown. Length of the part of cylinder in air is 'h' & lengths of part of cylinder immersed in the liquid are ${\text{h}}_1\&{\text{h}}_2$ as shown.                            

$\text{mg}={\text{F}}_{\text{b}}$  due to liquid (1) +${\text{F}}_{\text{b}}$  due to liquid (2)

$\text{v}\rho \text{g}={\text{v}}_1{\rho }_1\text{g}+{\text{v}}_{\text{2}}{\rho }_2\text{g}$

$\text{v}\rho ={\text{v}}_1{\rho }_1+v_2{\rho }_2$

${\text{Ah}}^{\text{1}}\rho ={\text{Ah}}_{\text{1}}{\rho }_1+{\text{Ah}}_{\text{2}}{\rho }_2$

${\text{h}}^{\text{1}}\rho ={\text{h}}_{\text{1}}{\rho }_1+{\text{h}}_{\text{2}}{\rho }_2$

$\left(\text{h}+{\text{h}}_{\text{1}}+{\text{h}}_{\text{2}}\right)\rho ={\text{h}}_{\text{1}}{\rho }_1+{\text{h}}_{\text{2}}{\rho }_2$

$\text{h}\rho +{\text{h}}_1\rho +{\text{h}}_{\text{2}}{\rho }_2={\text{h}}_{\text{1}}{\rho }_1+{\text{h}}_{\text{2}}{\rho }_2$

${\text{h}}_1\rho ={\text{h}}_1\left({\rho }_1-\rho \right)+{\text{h}}_{\text{2}}\left({\rho }_2-\rho \right)$

$\text{h}=\text{h}\left(\frac{{\rho }_1-\rho }{\rho }\right)+h_2\left(\frac{{\rho }_2-\rho }{\rho }\right)$

$\text{h}={\text{h}}_{\text{1}}\left(\frac{{\rho }_1}{\rho }-1\right)+{\text{h}}_{\text{2}}\left(\frac{{\rho }_2}{\rho }-1\right)$

When cylinder is depressed in such a way that its top surface is just covered by the liquid of density${\rho }_1$  and then released its height goes in liquid (2). So extra buoyant force due to liquid (2) acts on it. Due to this restoring force, cylinder is accelerated upward.

*Restoring force acting on the cylinder

$\text{F}=\text{hA}{\rho }_2\text{g}$

$=\text{hA}{\rho }_2\text{g}$

*Acceleration of the cylinder

$\text{a}=\frac{\text{F}}{\text{m}}=\frac{\text{hA}{\rho }_2\text{g}}{\left(\text{h}+{\text{h}}_{\text{1}}+{\text{h}}_{\text{2}}\right)}$

$=\frac{\text{h}{\rho }_2\text{g}}{\left(\text{h}+{\text{h}}_{\text{1}}+{\text{h}}_{\text{2}}\right)\rho }$